Precise UAV Navigation with Cellular Carrier Phase ... · Second, cellular carrier phase measurements are modeled at a fine granularity level to consist of four terms: true range,

Precise UAV Navigation with Cellular

Carrier Phase Measurements

Joe Khalife and Zaher M. Kassas

Department of Electrical and Computer Engineering

University of California, Riverside

Riverside, U.S.A.

[email protected], [email protected]

Abstract—This paper presents two frameworks for preciseunmanned aerial vehicle (UAV) navigation with cellular carrierphase measurements. The first framework relies on a mappingUAV and a navigating UAV sharing carrier phase measurements.Experimental results show that a 63.06 cm position root mean-square error (RMSE) is achieved with this framework. Thesecond framework leverages the relative stability of cellular basetransceiver station (BTS) clocks, which alleviates the need ofa mapper. It was shown through experimental data that thebeat frequency stability of cellular BTSs approaches that ofatomic standards and may be exploited for precise navigationwith cellular carrier phase measurements. Performance boundsare derived for this framework. Experimental data demonstratea single UAV navigating with sub-meter level accuracy for morethan 5 minutes, with one experiment showing 36.61 cm positionRMSE and another showing 88.58 cm position RMSE.

I. INTRODUCTION

Unmanned aerial vehicles (UAVs) will demand a resilient,

accurate, and tamper-proof navigation system. Current UAV

navigation systems will not meet these stringent demands

as they are dependent on global navigation satellite system

(GNSS) signals, which are jammable, spoofable, and may not

be usable in certain environments (e.g., indoors and deep urban

canyons) [1]–[3].

The potential of signals of opportunity (SOPs) (e.g.,

AM/FM radio [4], [5], iridium satellite signals [6], [7], WiFi

[8], [9], and cellular [10]–[13]) as alternative navigation

sources have been the subject of extensive research recently.

Navigation with SOPs has been demonstrated on ground

vehicles and UAVs, achieving a localization accuracy rang-

ing from meters to tens of meters, with the latter accuracy

corresponding to ground vehicles in deep urban canyons with

severe multipath conditions [14]–[17].

Cellular signals, particularly code-division multiple access

(CDMA) and long term evolution (LTE), are among the most

attractive SOP candidates for navigation. These signals are

abundant, received at a much higher power than GNSS signals,

offer a favorable horizontal geometry, and are free to use.

Several receiver designs have been proposed recently that

produce navigation observables from cellular CDMA and LTE

signals [18]–[21]. Moreover, error sources pertaining to code

phase-based navigation with cellular CDMA systems have

This work was supported in part by the Office of Naval Research (ONR)under Grant N00014-16-1-2305.

been derived and performance under such errors has been

characterized [13].

A different challenge that arises in cellular-based navigation

is the unknown states of the cellular base transceiver stations

(BTSs), namely their position and clock errors (bias and drift).

This is in sharp contrast to GNSS-based navigation where

the states of the satellites are transmitted to the receiver in

the navigation message. To deal with this challenge, a map-

per/navigator framework was proposed in [13], [20], where

the mapper was assumed to have complete knowledge of its

states (e.g., by having access to GNSS signals), estimating

the states of BTSs in its environment, and sharing these

estimates with a navigator that had no knowledge of its states,

making pseudorange measurements on the same BTSs in the

environment [13], [20]. Another framework was presented in

which the navigator estimated its states simultaneously with

the states of the BTSs in the environment, i.e., performed

radio simultaneous localization and mapping (radio SLAM)

[22]–[26]. It is worth noting that since cellular BTSs are

spatially stationary, their positions may be mapped prior to

navigation (e.g., by dedicated mapping receivers [27] or from

satellite imagery and cellular databases). However, the BTSs’

clocks errors must be continuously estimated, whether in the

mapper/navigator framework or radio SLAM framework, since

these errors are stochastic and dynamic.

The relative stability of cellular CDMA BTSs clocks was

recently studied, revealing that while these clocks are not

perfectly synchronized to GPS, the clock biases of differ-

ent neighboring BTSs are dominated by one common term

[28]. Moreover, experimental data recorded over 24-hours

showed that deviations from this common term are stable

processes. These key findings suggest that precise carrier

phase navigation with cellular signals is achievable with

or without a mapper. This paper presents a comprehensive

framework for precise UAV navigation using cellular carrier

phase measurements. Experimental results with the proposed

framework are presented, demonstrating sub-meter level UAV

navigation accuracy. Another contributing factor for achieving

such results is that cellular signals received by UAVs do not

suffer from severe multipath by virtue of the favorable channel

between cellular BTSs and UAVs. This can be seen in the clean

correlation functions calculated by the receiver. These results

are, to the authors’ knowledge, the most accurate navigation

results with cellular signals in the published literature. This

paper makes three contributions, which are discussed next.

First, two navigation frameworks for precise UAV naviga-

tion with cellular carrier phase measurements are developed.

The first framework consists of a mapping UAV and a navigat-

ing UAV that utilizes carrier phase differential cellular (CD-

cellular) measurements. The second framework consists of a

single navigating UAV, leveraging the relative stability of the

BTS clocks, estimating its position with a weighted nonlinear

least-squares (WNLS) estimator.

Second, cellular carrier phase measurements are modeled

at a fine granularity level to consist of four terms: true range,

common clock error, deviation from the common clock error,

and measurement noise. The deviation term is demonstrated to

evolve as a stable stochastic process, which is characterized via

system identification. Moreover, experimental results over long

periods of time validating the identified models are presented.

The paper also discusses how to estimate the statistics of

this process on-the-fly when the receiver has access to GNSS

signals.

Third, the navigation performance for the second framework

(single navigating UAV) is characterized. A theoretical lower

bound for the logarithm of the determinant of the position

estimation error covariance is derived and an upper bound on

the position error is provided.

Experimental results are provided demonstrating each of the

proposed frameworks. Two sets of experiments are performed

where sub-meter level UAV navigation with cellular carrier

phase signals is achieved for periods of over five minutes.

The remainder of the paper is organized as follows. Section

II describes the cellular carrier phase observable. Section

III describes the mapper/navigator framework. Section IV

describes the single UAV navigation framework that leverages

the relative stability of cellular SOPs. Section V derives

stochastic models for the clock deviations and validates these

models experimentally. Section VI establishes performance

bounds for the second proposed framework. Section VII

provides experimental results demonstrating each framework,

showing sub-meter level UAV navigation accuracy. Conclud-

ing remarks are given in Section VIII.

II. CELLULAR CARRIER PHASE OBSERVABLE MODEL

In cellular systems, several known signals may be trans-

mitted for synchronization or channel estimation purposes. In

CDMA systems, a pilot signal consisting of a pseudorandom

noise (PRN) sequence, known as the short code, is modulated

by a carrier signal and broadcast by each BTS for synchro-

nization purposes [29]. Therefore, by knowing the shortcode,

the receiver may measure the code phase of the pilot signal

as well as its carrier phase, hence forming a pseudorange

measurement to each BTS transmitting the pilot signal. In

LTE, two synchronization signals (primary synchronization

signal (PSS) and secondary synchronization signal (SSS)) are

broadcast by each evolved node B (eNodeB) [30]. In addition

to the PSS and SSS, a reference signal known as the cell-

specific reference signal (CRS) is transmitted by each eNodeB

for channel estimation purposes [30]. The PSS, SSS, and

CRS may be exploited to draw carrier phase and pseudorange

measurements on neighboring eNodeBs [21], [31]. In the rest

of this paper, availability of code phase and Doppler frequency

measurements of cellular CDMA and LTE signals is assumed

(e.g., from specialized navigation receivers [19] [20] [12].

The continuous-time carrier phase observable can be ob-

tained by integrating the Doppler measurement over time [32].

The carrier phase (expressed in cycles) made by the i-threceiver on the n-th SOP is given by

φ(i)n (t) = φ(i)n (t0) +

∫ t

t0

f(i)Dn

(τ)dτ, n = 1, . . . , N, (1)

where f(i)Dn

is the Doppler measurement made by the i-

th receiver on the n-th cellular SOP, φ(i)n (t0) is the initial

carrier phase, and N is the total number of SOPs. In (1), idenotes either the mapper M or the navigator N. Assuming a

constant Doppler during a subaccumulation period T , (1) can

be discretized to yield

φ(i)n (tk) = φ(i)n (t0) +

k−1∑

l=0

f(i)Dn

(tl)T, (2)

where tk , t0 + kT . In what follows, the time argument tkwill be replaced by k for simplicity of notation. Note that the

receiver will make noisy carrier phase measurements. Adding

measurement noise to (2) and expressing the carrier phase

observable in meters yields

z(i)n (k) = λφ(i)n + λT

k−1∑

l=0

f(i)Dn

(l) + v(i)n (k), (3)

where λ is the wavelength of the carrier signal and v(i)n is the

measurement noise, which is modeled as a discrete-time zero-

mean white Gaussian sequence with variance(

σ(i)n

)2

, which

can be shown for a coherent second-order phase lock loop

(PLL) to be given by [32]

(

σ(i)n

)2

= λ2Bi,PLL

C/N0n

,

where Bi,PLL is the receiver’s PLL noise equivalent bandwidth

and C/N0n is the cellular SOP’s measured carrier-to-noise

ratio. Note that a coherent PLL may be employed in CDMA

and LTE navigation receivers since the cellular synchroniza-

tion and reference signals do not carry any data. The carrier

phase in (3) can be parameterized in terms of the receiver and

cellular SOP states as

z(i)n (k) = ‖rri(k)− rsn‖2+ c [δtri(k)− δtsn(k)]

+ λN (i)n + v(i)n (k), (4)

where rri , [xri , yri ]T

is the receiver’s position vector;

rsn , [xsn , ysn ]T

is the cellular BTS’s position vector; c is

the speed of light; δtri and δtsn are the receiver’s and cellular

BTS’s clock biases, respectively; and N(i)n is the carrier phase

ambiguity. Note that the difference between the UAV’s height

and the cellular BTSs’ heights is typically negligible compared

to the range between the UAV and the BTSs. Therefore, one

may estimate the UAV’s two-dimensional (2–D) position only,

without introducing significant errors. Moreover, an altimeter

may be used to estimate the UAV’s height. The subsequent

analysis may be readily extended to 3–D; however, the vertical

position estimate will suffer from large uncertainties due to the

poor vertical diversity of cellular SOPs.

III. NAVIGATION WITH SOP CARRIER PHASE

DIFFERENTIAL CELLULAR MEASUREMENTS

In this section, a framework for CD-cellular navigation is

developed. The framework consists of two UAVs in an envi-

ronment comprising N cellular BTSs. The UAVs are assumed

to be listening to the same BTSs with the BTS locations

being known. The first UAV, referred to as the mapper (M), is

assumed to have knowledge of its own position state (e.g., a

high-flying UAV with access to GNSS or one equipped with

a sophisticated sensor suite). Note that instead of a UAV, the

mapper may be a stationary receiver deployed at a surveyed

location. The second UAV, referred to as the navigator (N),

is assumed only to know its position between k = 0 and

k = k0 (e.g., from GNSS). For k > k0, the navigator is

supposed to subsequently navigate exclusively with cellular

carrier phase observables (e.g., its access to GNSS signals

was cut off). The mapper communicates its own position and

carrier phase observables with the navigator. Fig. 1 illustrates

the mapper/navigator framework.

Central

BTS n

Database

BTS 2

BTS 1

Mapper

xsn; ysn

Data: xrM; yrM ;

{

z(M)n ;

(

σ(M)n

)2}N

n=1

Data

Navigator

(Estimating xrNand yrN)

Fig. 1. Mapper/navigator framework.

In what follows, the objective is to estimate the navigator’s

position, which will be achieved by double-differencing the

measurements (4). Without loss of generality, let the measure-

ments to the first SOP be taken as references to form the single

difference

z(i)n,1(k) , z(i)n (k)− z

(i)1 (k).

Subsequently, define the double difference between N and Mas

z(N,M)n,1 (k) , z

(N)n,1 (k)− z

(M)n,1 (k)

+ ‖rrM(k)− rsn‖2 − ‖rrM(k)− rs1‖2

, h(N)n,1 (k) + λN

(N,M)n,1 + v

(N,M)n,1 (k), (5)

where n = 1, . . . , N , hn,1(N)(k) , ‖rrN(k)− rsn‖2 −

‖rrN(k)− rs1‖2, N(N,M)n,1 , N

(N)n −N

(M)n −N

(N)1 +N

(M)1 , and

v(N,M)n,1 (k) , v

(N)n (k)− v

(M)n (k) − v

(N)1 (k) + v

(M)1 (k). Define

the state to be estimated as the navigator’s position x , rrNand the vector of measurements as

z(k) , h [x(k)] + λN + v(k),

where

z(k) ,[

z(N,M)2,1 (k), . . . , z

(N,M)M,1 (k)

]T

h [x(k)] ,[

h(N)2,1 (k), . . . , h

(N)M,1(k)

]T

N ,

[

N(N,M)2,1 , . . . , N

(N,M)M,1

]T

v(k) ,[

v(N,M)2,1 (k), . . . , v

(N,M)M,1 (k)

]T

,

where v(k) has a covariance RN,M which can be readily

shown to be

RN,M = R(1) +

[

(

σ(M)1

)2

+(

σ(N)1

)2]

Ξ,

where

R(1),diag

[

(

σ(M)2

)2

+(

σ(N)2

)2

, . . . ,(

σ(M)N

)2

+(

σ(N)N

)2]

and Ξ is a matrix of ones. Note that the vector N is now

a vector of integers and has to be known to solve for the

navigator’s position. To this end, the mapper and navigator will

leverage the period where they both know their positions to

solve for N through an integer least-squares (ILS) estimator.

Define y(k) to be

y(k) ,1

λ{z(k)− h [x(k)]} = N +

1

λv(k).

Using all measurements {y(k)}k0

k=0, one may solve for the

float solution of N either using a batch weighted least-squares

(LS) estimator or using a recursive LS estimator. Then a

decorrelation method can be used to solve for the integer parts.

This will yield the estimate N and an associated estimation

error of δN such that N = N+δN with the estimation error

covariance QN [33]. For k > k0, define the new measurement

z′(k) according to

z′(k) , z(k)− λN = h [x(k)] + v′(k),

where v′(k) , v(k)+λδN is a zero-mean random vector with

covariance R′

N,M = RN,M+λ2QN and k > k0. Subsequently,

one may solve for x(k) where k > k0 using a weighted

nonlinear LS (WNLS) estimator.

IV. NAVIGATION WITH SOP CARRIER PHASE

MEASUREMENTS: SINGLE UAV

The mapper/navigator framework presented above requires

the presence of a mapper and a communication channel

between the mapper and the navigator. This section discusses a

cellular carrier phase navigation framework that alleviates the

need of a mapper, i.e., employable on a single UAV. Note that

since what follows only pertains to single UAV navigation, the

UAV index i will be dropped for simplicity of notation.

The terms c[

δtr(k)− δtsn(k) +λcNn

]

are combined into

one term defined as

cδtn(k) , c

[

δtr(k)− δtsn(k) +λ

cNn

]

.

It was noted in [28] that cellular BTSs possess tighter carrier

frequency synchronization then time (code phase) synchro-

nization (the code phase synchronization requirement as per

the cellular protocol is to be within 3 µs). Therefore, the

resulting clock biases in the carrier phase estimates will be

very similar, up to an initial bias, as shown in Fig. 2. Conse-

quently, one may leverage this relative frequency stability to

eliminate parameters that need to be estimated. Moreover, this

allows one to use a static estimator (e.g., a WNLS) to estimate

the position of the UAV. To achieve this, in what follows,

the carrier phase measurement is first re-parameterized and a

WNLS estimation framework is subsequently developed.

0 2 4 6 8 10 12 14 16 18 20 22 24-150

-100

-50

0

50

Fig. 2. Experimental data showing cδtn(k)− cδtn(0) obtained from carrierphase measurements over 24 hours for three neighboring BTSs. It can beseen that the clock biases cδtn(k) in the carrier phase measurement are verysimilar up to an initial bias cδtn(0) which has been removed.

A. Carrier Phase Measurement Re-Parametrization

Motivated by the experimental results in [28], the following

re-parametrization is proposed

cδtn(k) , cδtn(k)− cδtn(0) ≡ cδt(k) + ǫn(k), (6)

where cδt is a time-varying common bias term and ǫn is the

deviation of cδtn from this common bias and is treated as

measurement noise. Using (6), the carrier phase measurement

(4) can be re-parameterized as

zn(k) = ‖rr(k)− rsn‖2 + cδt(k) + cδt0n + ηn(k), (7)

where cδt0n , cδtn(0) and ηn(k) , ǫn(k)+vn(k) is the over-

all measurement noise. The statistics of ǫn will be discussed in

Section V. Note that cδt0n can be obtained knowing the initial

position and given the initial measurement zn(0) according

to cδt0n ≈ zn(0) − ‖rr(0)− rsn‖2. This approximation

ignores the contribution of the initial measurement noise. If

the receiver is initially stationary for a period k0T seconds,

which is short enough such that δt(k) ≈ 0 for k = 1, . . . , k0,

then the first k0 samples may be averaged to obtain a more

accurate estimate of cδt0n .

It is proposed that instead of lumping all N clock biases

into one bias cδt to be estimated, several clusters of clocks

get formed, each of size Nl (i.e.,L∑

l=1

Nl = N , where L is the

total number of clusters), and the clocks in each cluster are

lumped into one bias cδtl to be estimated. This gives finer

granularity for the parametrization (6), since naturally, certain

groups of cellular SOPs will be more synchronized with each

other than with other groups (e.g., corresponding to the same

network provider, transmission protocol, etc.). An illustrative

experimental plot is shown in Fig. 3. Note that since the 2–D

position vector of the UAV is being estimated along with Lclock biases, the number of clusters L cannot exceed N − 2,

otherwise there would be more unknowns than measurements.

100 105 110 115 120 125 1306

8

10

12

14

Fig. 3. Experimental data for cδtn(k) over 30 seconds for 8 BTSs. Theclock biases have been visually clustered into three clusters as an illustrativeexample.

Without loss of generality, it assumed that the carrier

phase measurements have been ordered such that the first N1

measurements were grouped into the first cluser, the second

N2 measurements were grouped into the second cluster, and

so on. Next, obtaining the navigation solution with a WNLS

is discussed.

B. Navigation Solution

Given N ≥ 3 pseudoranges modeled according to (7) and

L ≤ N − 2 SOP clusters, the receiver may solve for its

current position rr and the current set of common biases

cδt , [cδt1, . . . , cδtL]T

using a WNLS estimator. The state

to be estimated is defined by x ,[

rT

r , cδtT]T

. An estimate

x may be obtained using the iterated WNLS equations given

by

x(j+1)(k) = x

(j)(k) +(

HTR−1η H

)−1HTR−1

η δz(k), (8)

where δz(k) , [δz1(k), . . . , δzN (k)]T and δzn(k) ,

zn(k) −[∥

∥

∥r(j)r (k)− rsn

∥

∥

∥

2+ cδt

(j)ln

(k) + cδt0n

]

, Rη =

diag[

σ21 + σ2

ǫ1, . . . , σ2

N + σ2ǫN

]

is the measurement noise co-

variance where σ2ǫn

will be discussed in Section V, j is the

WNLS iteration index, and H is the measurement Jacobian

given by

H , [G Γ] , Γ ,

1N1. . . 0

.... . .

...

0 . . . 1NL

, (9)

G ,

r(j)r − rs1

∥

∥

∥r(j) − rs1

∥

∥

∥

2

. . .r(j)r − rsN

∥

∥

∥r(j) − rsN

∥

∥

∥

2

T

, (10)

and 1Nl, [1, . . . , 1]

T. Note that

ln =

1, for n = 1, . . . , N1,

2, for n = N1 + 1, . . . ,∑2

l=1Nl,...

...

L, for n =∑L−1

l=1 Nl + 1, . . . , N.

After convergence (i.e., x(j+1)(k) ≈ x(j)(k)) the final esti-

mate is obtained by setting x(k) ≡ x(j+1)(k). In the rest of

the paper, it is assumed that H is always full column rank.

C. Common Clock Bias Parametrization

Note that the clock bias clusters {cδtl}Ll=1 are “virtual

clock biases”, which are introduced to group SOPs whose

carrier frequency is more synchronized than others. This would

in turn yield more precise measurement models, reducing

the estimation error. This subsection parameterizes cδtl as

a function of cδtn. This parametrization is based on the

following theorem.

Theorem IV.1. Consider N ≥ 3 carrier phase measurements.

Assume that the contribution of the relative clock deviation ǫnis much larger than the carrier phase measurement noise vnand that ǫn are uncorrelated with identical variances σ2. Then,

the position error at any time instant δrr(k) due to relative

clock deviations is independent of cδtl.

Proof. Denote the measurement noise covariance of η ,

[η1 . . . , ηn]T

as Rη. It is assumed that the WNLS had con-

verged very closely to the true state in the absence of clock

deviations. The clock deviations are then suddenly introduced

into the measurements, which will induce an incremental

change in the receiver state estimate given by

δx(k) = −(

HTR−1η H

)−1HTR−1

η ǫ(k)

= −(

HTH)−1

HTǫ(k),

where

H , R−

12

η H, ǫ(k) , R−

12

η ǫ(k),

and ǫ , [ǫ1, . . . , ǫN ]T

. The matrix product HTǫ(k) can be

further expressed as

HTǫ(k) =

[

GT

ΓT

]

ǫ(k) =

[

GTǫ(k)ΓTǫ(k)

]

,

where

G , R−

12

η G, Γ , R−

12

η Γ.

Next,(

HTH)

−1is expressed as

(

HTH)−1

=

[

GTG GTΓ

ΓTG ΓTΓ

]−1

,

[

A B

BT D

]

,

where A is a 2 × 2 symmetric matrix, B is a 2 × L matrix,

and D is an L × L symmetric matrix. The estimation error

becomes

δx(k) =

[

δrr(k)δ (cδt(k))

]

= −

[ (

AGT +BΓT)

ǫ(k)(

BTGT +DΓT)

ǫ(k)

]

.

Using the matrix block inversion lemma, the following may

be obtained

A =(

GTΨG)−1

B = −(

GTΨG)−1

GTΓ(

ΓTΓ)−1

D =(

ΓTΓ)−1

[

I+ ΓTG(

GTΨG)−1

GTΓ(

ΓTΓ)−1

]

,

where Ψ , I− Γ(

ΓTΓ)

−1ΓT. This yields the position error

given by

δrr(k) = −(

GTΨG)−1

GTΨǫ(k).

When Rη = σ2I, the above simplifies to

δrr(k) = −(

GTΨG)−1

GTΨǫ(k), (11)

ǫ(k) , [ǫ1(k), . . . , ǫN (k)]T , Ψ , I− Γ(

ΓTΓ)−1

ΓT.(12)

Note that Ψ is the annihilator matrix of Γ and satisfies ΨΨ =Ψ. It can be readily shown that

Ψ = diag

[

IN1−

1

N11N1

1T

N1, . . . , INL

−1

NL

1NL1T

NL

]

.

Consequently, (11) implies that the effect on the position error

δrr comes from the vector

ǫ(k) , Ψǫ(k) = −

ǫ1(k)− µ1(k)1N1

...

ǫL(k)− µL(k)1NL

,

where ǫ(k) =[

ǫT1 (k), . . . , ǫT

L(k)]T

, ǫl(k) =[

ǫl1 , . . . , ǫlNl

]T

,

and µl(k) ,1Nl

∑Nl

i=1 ǫli(k). Noting that ǫn(k) = cδtln(k)−

cδtn(k), the following holds

ǫn(k) =1

Nl

Nl∑

i=1

[

cδtl(k)− cδtli(k)]

−[

cδtl(k)− cδtn(k)]

= cδtn(k)−1

Nl

Nl∑

i=1

cδtli(k), (13)

which is independent of cδtl(k).

The assumption that the contribution of the relative clock

deviation ǫn is much larger than the carrier phase measurement

noise vn comes from experimental data, where ‖ǫ‖2 was

observed to be within 0.2 and 4 m, whereas σn was on the

order of a few cm. Form Theorem IV.1, it can be implied

that while the position error is independent of cδtl, it depends

on the clustering. Following the result in (13), the following

parametrization is adopted

cδtl(k) ≡1

Nl

Nl∑

i=1

cδtli(k), ǫn(k) ≡ cδtn(k)− cδtl(k).

(14)

The following section models the dynamics of ǫn.

V. FREQUENCY STABILITY AND MODELING THE

DYNAMICS OF CLOCK DEVIATIONS

In this section, the frequency stability and the deviations ǫnin cellular CDMA systems are characterized.

A. Observed Frequency Stability in Cellular CDMA Systems

In order to study the stability of cellular CDMA BTS clocks,

real CDMA signals were collected over a period of 24 hours

via a stationary universal software radio peripheral (USRP)

driven by a GPS-disciplined oscillator (GPSDO). Since the

USRP clock is driven by a GPSDO, the apparent Doppler

frequency will be mainly caused by the drift in the BTS

clock. The Allan deviations were calculated for each BTS

using: (1) the absolute Doppler frequencies and (2) the beat

Doppler frequencies. The absolute Doppler frequencies are the

frequencies directly observed by the receiver on each BTS.

The beat Doppler frequency is defined as

fbDn, fDn

−1

N

N∑

n=1

fDn,

following the parametrization in (14). The Allan deviations of

the absolute and beat frequencies for three cellular CDMA

BTSs nearby the campus of the University of California,

Riverside (UCR) are shown in Fig. 4. Note that the absolute

and beat Doppler frequencies were normalized by the nominal

carrier frequency fc; hence, the Allan deviations are unitless.

10-2

100

102

104

10-12

10-11

10-10

10-9

*°

Fig. 4. Allan deviations of absolute and beat frequencies for three CDMABTSs near UCR. The Allan deviations were calculated from data collectedover 24 hours. The carrier frequency was fc = 883.98 MHz.

Two main conclusions may be drawn from Fig. 4. First,

the beat frequencies are an order of magnitude more stable

than the absolute frequencies. Second, the stability of the beat

frequencies approaches that of atomic standards for periods of

hundreds to a few thousands seconds. This implies that cellular

CDMA signals may be exploited for precise navigation for

several minutes using carrier phase measurements.

A similar experiment was conducted at a different time in

Colton, California. However, only ten minutes of data were

collected. The Allan deviations for two cellular CDMA BTSs

in Colton, California, are shown in Fig. 5. Similar conclusions

are drawn from Fig. 5.

10-2

100

102

104

10-12

10-11

10-10

10-9

10-8

°

Fig. 5. Allan deviations of absolute and beat frequencies for two CDMABTSs in Colton, California. The Allan deviations were calculated from datacollected over ten minutes. The carrier frequency was fc = 882.75 MHz.

B. Modeling the Dynamics of Clock Deviations

Fig. 6 shows the clock bias deviations {ǫn}3n=1 for the three

cellular BTSs nearby UCR over 24 hours.

0 2 4 6 8 10 12 14 16 18 20 22 24-10

-5

0

5

10

Fig. 6. Plot of the deviations ǫn(k) from the common clock bias for threeBTSs near UCR over 24 hours.

The UAV can perform an exhaustive search over the dif-

ferent clustering possibilities to minimize its position error

while it has access to GPS. The number of possible clusters

is given by Nclus =N−2∑

L=1

(

NL

)

=N−2∑

L=1

N !L!(N−L)! . It can

be seen that this number becomes impractically large as Nincreases. A rule-of-thumb that significantly reduces Nnum

is discussed in Subsection VI-C. Subsequently, it is assumed

that a clustering is given. Next, ǫn are calculated according to

(14). It can be seen from Fig. 6 that ǫn is bounded. It can be

readily verified (e.g., through spectral analysis) that ǫn is not a

white sequence. An auto regressive moving average (ARMA)

model is proposed to describe the dynamics of ǫn, which is

generically expressed as

ǫn(k+1) =

p∑

i=1

φiǫn(k−i+1)+

q∑

i=1

ψiwǫn(k−i+1)+wǫn(k),

(15)

where p and {φi}pi=1 are the order and the coefficients of the

autoregressive (AR) part, respectively; q and {ψi}q

i=1 are the

order and the coefficients of the moving average (MA) part,

respectively; and wǫ is a white sequence. Identifying p and qand their corresponding coefficients can be readily obtained

with standard system identification techniques [34]. Here, the

MATLAB System Identification Toolbox was used to identify

(15), where it was found that p = q = 6 was usually enough

to whiten wǫn .

C. Statistics of the Residuals

In this subsection, the resulting residuals wǫ are studied. To

this end, the autocorrelation function (acf) and the probability

density function (pdf) of the residuals are computed for the

three realizations of ǫn shown in Fig. 6. Note that half of the

data was used for system identification and the other half was

used to validate the model. The acf and pdf of the residuals

obtained with the second half of the data are plotted in Figs.

7(a)–(c). A Gaussian pdf fit (red) was also plotted. It can be

seen that {wǫn}3n=1 are zero-mean white Gaussian sequences,

with variances{

σ2wǫn

}3

n=1.

Fig. 7. (a), (b), and (c) show the acfs and pdfs of wǫ1, wǫ2

, and wǫ3,

respectively. The acfs show that the sequences {wǫn}3

n=1are approximately

white and the pdfs show that the sequences are Gaussian.

D. Statistics of the Clock Deviations

Since wǫn(k) was found to be a Gaussian sequence, then ǫn,

which is a linear combination of wǫn(k) will also be Gaussian.

Without loss of generality, it is assumed that ǫn(i − p) = 0for i = 1, . . . , p. Subsequently, E [ǫn(k)] = 0. The variance of

ǫn(k) is discussed next. The ARMA process identified earlier

may be represented in state-space according to

ξn(k + 1) = Fξnξn(k) + Γξnwǫn(k)

ǫn(k) = hT

ǫnξn(k)

where ξn is the underlying dynamic AR process, Fξn is its

state transition matrix, Γξn is the input matrix, and hT

ǫnis the

output matrix. The eigenvalues of Fξn were computed to be

inside the unit circle, implying stability of ξn. The covariance

of ξn, denoted Pξn , evolves according to

Pξn(k + 1) = FξnPξn(k)FT

ξn+Qξn ,

where Qξn , σ2wǫn

ΓξnΓT

ξnand the variance of the clock

deviation ǫn at any given time-step is given by

σ2ǫn(k) = hT

ǫnPξn(k)hǫn .

Since ξn is stable, Pξn(k) will converge to a finite steady-

state covariance denoted Pξn,ss given by the solution to the

discrete-time matrix Lyapunov equation

Pξn,ss = FξnPξn,ssFT

ξn+Qξn .

Subsequently, the steady-state variance of the clock deviation

is given byσ2ǫn

= hT

ǫnPξn,sshǫn .

VI. PERFORMANCE CHARACTERIZATION

This section derives performance bounds for the single UAV

navigation framework using SOP carrier phase measurements

presented in Section IV. Also, clustering of the clock bias

biases is investigated and an upper bound on the position error

is derived.

A. A Note on the Optimal BTS Geometric Configuration

The measurement Jacobian G with respect to the position

states (cf. (10)) could be re-parameterized in terms of the

bearing angles θn between each SOP and the UAV, given by

G =

[

cos θ1 . . . cos θNsin θ1 . . . sin θN

]T

,

as illustrated in Fig. 8(a). The optimal geometric configuration

of sensors (or navigation sources) around an emitter (or

receiver) has been well studied in the literature. This problem

is also similar to the geometric dilution of precision (GDOP)

minimization problem in GPS. It was found that the GDOP is

minimized when the end points of the unit line of sight vectors

pointing from the receiver to each navigation source form a

regular polygon around the receiver, as shown in Fig. 8(b). In

the sequel, the aforementioned configuration will be referred

to as the optimal configuration, where the bearing angles are

given by θn = 2π(n−1)N

, n = 1, . . . , N . Note that these results

hold for N ≥ 3 in the 2-D case.

BTS 1

θ1

θ2

x

y

1

BTS 2

x

y

(a) (b)

BTS 3

θ3

Fig. 8. (a) Re-parametrization of the measurement Jacobian as a functionof the bearing angles θn. (b) Optimal geometric configuration of the BTSsaround the receiver.

B. Lower Bound on the logarithm of the Determinant of the

Position Estimation Error Covariance

It can be readily seen that optimal performance is achieved

when all clocks are perfectly synchronized, i.e., ǫn(k) = 0,

∀ k, and therefore Rη = R. In this case, only one clock

bias is estimated, and this problem becomes similar to the

one discussed in [13], in which it is shown that the logarithm

of the determinant of the position estimation error covariance

Px,y is bounded by

log det [Px,y] ≥ −2 log[

trace(

R−1)]

.

C. Clustering of the Clock Biases

It was mentioned in Subsection V-B that an exhaustive

search may be performed to cluster the clock biases cδtn in

order to minimize the position estimation error. This amounts

to finding the matrix Γ that minimizes

Jp(Γ) ,

k0∑

k=1

‖δrr(k)‖22 =

k0∑

k=1

∥

∥

∥

∥

[

GT

(

I−Γ(

ΓTΓ)−1

ΓT)

G]

−1

GT

(

I−Γ(

ΓTΓ)−1

ΓT)

ǫ(k)

∥

∥

∥

∥

2

2

=

k0∑

k=1

∥

∥

∥

(

GTΨG)−1

GTΨǫ(k)∥

∥

∥

2

2,

where Γ and Ψ are defined in (9) and (12), respectively. This

optimization problem is non-convex and intractable. Instead of

optimizing Jp(Γ), a tractable rule-of-thumb is provided next.

First, consider the modified cost function

J(Γ) ,∥

∥

∥

(

GTΨG)−1

GTΨǫ(k0)∥

∥

∥

2

2

=∥

∥

∥

(

GTΨΨG)−1

GTΨΨǫ(k0)∥

∥

∥

2

2

≤∥

∥

∥

(

GT

ΓGΓ

)−1GT

Γ

∥

∥

∥

2

2‖Ψǫ(k0)‖

22 ,

where GΓ , ΨG. Let the singular value decomposition (svd)

of GΓ be

GΓ = UΣΓVT,

where U is an N × N unitary matrix, V is a 2 × 2 unitary

matrix, and ΣΓ = [Σ 0]T

, where Σ is a nonsingular 2 × 2diagonal matrix containing the nonzero singular values of GΓ.

It can be readily shown that(

GT

ΓGΓ

)−1GT

Γ = VΣ′UT, (16)

where Σ′ ,[

Σ−1 0]T

. This implies that (16) is the svd of(

GT

ΓGΓ

)

−1GT

Γ and its singular values are the inverses of the

singular values of GΓ, yielding

∥

∥

∥

(

GT

ΓGΓ

)−1GT

Γ

∥

∥

∥

2

2= [σmax (GΓ)]

2=

[

1

σmin (GΓ)

]2

,

where σmax (·) and σmin (·) denote the maximum and mini-

mum singular values of a matrix, respectively. Note that the

singular values of GΓ are the square root of the eigenvalues

of GT

ΓGΓ = GTΨG, and hence∥

∥

∥

(

GT

ΓGΓ

)−1GT

Γ

∥

∥

∥

2

2=

1

λmin (GTΨG)= λmax (Px,y) ,

where λmax (·) and λmin (·) denote the maximum and mini-

mum eigenvalues of a matrix, respectively. Consequently, the

cost J(Γ) may be bounded by

J(Γ) ≤ λmax (Px,y) ‖Ψǫ(k0)‖22 . (17)

Next, two theorems are presented that will help derive the

rule-of-thumb for clustering the clock biases.

Theorem VI.1. Assume a clock bias clustering with L < N−2 clusters and denote JL , ‖Ψǫ(k)‖22. Then, there exists a

clustering with L+ 1 clusters such that JL ≥ JL+1.

Proof. First, note that JL may be expressed as

JL = ‖Ψǫ(k)‖22 =

∥

∥

∥

∥

∥

∥

∥

ǫ1(k)− µ1(k)1N1

...

ǫL(k)− µL(k)1NL

∥

∥

∥

∥

∥

∥

∥

2

2

=L∑

l=1

‖ǫl − µl(k)1Nl‖22 =

L∑

l=1

Nl∑

j=1

[

ǫlj (k)− µl(k)]2

=

L−1∑

l=1

Nl∑

j=1

[

ǫlj (k)− µl(k)]2

+

NL∑

j=1

[

ǫLj(k)− µl(k)

]2

= a+

NL∑

j=1

(ǫLj(k)− µL(k))

2,

where a ,∑L−1

l=1

∑Nl

j=1

[

ǫlj (k)− µl(k)]2

. In what follows,

the time argument k will be dropped for simplicity of notation.

Now add an additional cluster by partitioning ǫL according to

ǫL =[

ǫ′T

L, ǫL+1

]T

and define

JL+1 = a+

NL−1∑

j=1

(

ǫLj− µ′

L

)2+ (ǫL+1 − µL+1)

2,

where µ′

L , 1NL−1

NL−1∑

j=1

ǫLjand µL+1 = ǫL+1. Subsequently,

JL+1 may be expressed as

JL+1 = a+

NL−1∑

j=1

(ǫLj− µL)

2.

The second term in JL may be expressed as

NL∑

j=1

(ǫLj− µL)

2 =

NL∑

j=1

ǫ2Lj−NLµ

2L

=

NL−1∑

j=1

ǫ2Lj−NLµ

2L + ǫ2LNL

.

The term NLµ2L may be expressed as

NLµ2L = NL

1

NL

NL∑

j=1

ǫLj

2

=1

NL

NL−1∑

j=1

ǫLj+ ǫLNL

2

=1

NL

[

(NL − 1)µ′

L + ǫLNL

]2

=(NL − 1)

2µ′2

L

NL

+2(NL − 1)µ′

LǫLNL

NL

+ǫ2LNL

NL

= (NL − 1)µ′2L −

(NL − 1)µ′2L

NL

+2(NL − 1)µ′

LǫL,NL

NL

+ǫ2LNL

NL

+ ǫ2LNL− ǫ2LNL

= (NL − 1)µ′2L −

(NL − 1)

NL

(ǫLNL− µ′

L)2 + ǫ2LNL

.

Substituting back in the second term of JL yields

NL∑

j=1

(ǫLj−µL)

2 =

NL−1∑

j=1

(ǫLj−µ′

L)2+

(NL − 1)

NL

(ǫLNL−µ′

L)2.

Substituting back in JL yields

JL = a+

NL−1∑

j=1

(ǫLj− µ′

L)2 +

(NL − 1)

NL

(ǫLNL− µ′

L)2

= JL+1 +(NL − 1)

NL

(ǫLNL− µ′

L)2.

Since(NL−1)

NL(ǫLNL

− µ′

L)2 ≥ 0, then JL ≥ JL+1.

From Theorem VI.1, it can be implied that ‖Ψǫ(k)‖22 is

minimized when L = N − 2, i.e., the maximum number of

clusters is used. This also implies that using more SOP clusters

will decrease ‖Ψǫ(k0)‖22 in the upper bound expression of

J(Γ) given in (17).

Theorem VI.2. Consider N ≥ 3 carrier phase measurements

for estimating the receiver’s position rr and a clustering

of L clock states cδt. Adding a carrier phase measurement

from an additional cellular SOP while augmenting the clock

state vector cδt by its corresponding additional clock state

will neither change the position error nor the position error

uncertainty.

Proof. The augmented Jacobian matrix is given by

H′ =

[

G Γ 0

gT 0T 1

]

,

where g ,rr−rsN+1

‖rr−rsN+1‖2

. The new information matrix is

subsequently given by

H′TH′ =

GTG+ ggT GTΓ g

ΓTG ΓTΓ 0

gT 0T 1

=

[

M11 m12

mT

12 1

]

,

where

M11 ,

[

GTG+ ggT GTΓ

ΓTG ΓTΓ

]

,m12 ,

[

g

0

]

.

The new covariance is given by

P′ =(

H′TH′

)

−1

=

[

A′ b′

b′T

d′

]

,

where

A′ =(

M11 −m12mT

12

)−1

b′ = −(

M11 −m12mT

12

)−1m12

d′ = 1 +mT

12

(

M11 −m12mT

12

)−1m12

The matrix A′ may be expressed as

A′ =

([

GTG+ ggT GTΓ

ΓTG ΓTΓ

]

−

[

ggT 0

0T 0

])−1

=

[

GTG GTΓ

ΓTG ΓTΓ

]−1

= P,

which indicates that the new uncertainty in the position state

is unchanged. The new covariance can be expressed as

P′ =

[

P −Pm12

−mT

12P 1 +mT

12Pm12

]

=

P′

11 P′

12 P′

13

P′T

12 P′

22 P′

23

P′T

13 P′T

23 P′

33

,

where

P′

11 =(

GTΨG)−1

P′

12 = −(

GTΨG)−1

GTΓ(

ΓTΓ)−1

P′

13 = −(

GTΨG)−1

g

P′

22 =(

ΓTΓ)−1

ΓT

[

I+G(

GTΨG)−1

GT

]

Γ(

ΓTΓ)−1

P′

23 =(

ΓTΓ)−1

ΓTG(

GTΨG)−1

g

P′

33 = 1 + gT(

GTΨG)−1

g

The new estimation error is given by

δr′

r(k) = −P′H′Tǫ′(k),

where ǫ′(k) ,[

ǫT(k), ǫN+1(k)]T

and ǫN+1(k) is the error

from the (N+1)st measurement. Using the expressions of P′,

H′, and ǫ′, it can be readily shown that

δr′r(k) = −(

GTΨG)−1

GTΨǫ(k) = δrr(k).

Therefore, the addition of a measurement while augmenting

the clock state vector by one state will not improve the position

estimate nor the position error uncertainty.

From Theorem VI.2, it can be implied that it is required

that Nl ≥ 2 in order for cluster l to contribute in estimat-

ing the position state. Therefore, λmax (Px,y) can be made

smaller by decreasing the number of clusters L. Combining

the conclusions of Theorems VI.1 and VI.2 and referring to

(17), one can see that there is a tradeoff between estimating

more clock biases and uncertainty reduction: less bias for

more uncertainty and vice versa. Subsequently, a good rule

of thumb is to have at least on cluster with Nl ≥ 3 (to ensure

observability) and Nl ≥ 2 for the remaining clusters. This

implies that L ≤ N−32 + 1, which significantly reduces the

number of possible clusters in the exhaustive search algorithm.

D. Upper Bound on the Position Error

Note that for a given number of SOPs, one will choose

a clustering that will yield a performance that is at least as

good as estimating one clock bias. Therefore, a bound on the

position error may be established according to

‖δrr(k)‖2 ≤∥

∥

∥

(

GTΨ1G)−1

GTΨ1ǫ(k)∥

∥

∥

2,

where Ψ1 , I− 1N1N1T

N .

VII. EXPERIMENTAL RESULTS

In this section, experimental results are presented demon-

strating precise, sub-meter level UAV navigation results via

the two frameworks developed in this paper: (1) CD-cellular

with a mapper/navigator and (2) single UAV with precise

carrier phase measurements. As mentioned in Section III,

only the 2–D positions of the UAVs are estimated as their

height may be obtained using other sensors (e.g., altimeter).

In the following experiments, the height of the UAVs was

obtained from their on-board navigation systems. Moreover,

the noise equivalent bandwidth of the receivers’ PLL was set

to BN,PLL = BM,PLL = BPLL = 3 Hz in all experiments.

A. Carrier Phase Differential Cellular UAV Navigation Re-

sults via the Mapper/Navigator Framework

In order to demonstrate the mapper/navigator framework

discussed in Section III, two Autel Robotics X-Star Premium

UAVs were equipped each with an Ettus E312 USRP, a

consumer-grade 800/1900 MHz cellular antenna, and a small

consumer-grade GPS antenna to discipline the on-board os-

cillator. The receivers were tuned to a 882.75 MHz carrier

frequency (i.e., λ = 33.96 cm), which is a cellular CDMA

channel allocated for the U.S. cellular provider Verizon Wire-

less. Samples of the received signals were stored for off-

line post-processing. The cellular carrier phase measurements

were given at a rate of 37.5 Hz, i.e., T = 0.0267 ms. The

ground-truth reference for each UAV trajectory was taken

from its on-board navigation system, which uses GPS, an

inertial measurement unit (IMU), and other sensors. The

navigator’s total traversed trajectory was 1.72 Km, which was

completed in 3 minutes. Over the course of the experiment, the

receivers on-board the UAVs were listening to 9 BTSs, whose

positions were mapped prior to the experiment according to

the framework discussed in [27]. A plot of the carrier-to-noise

ratios of all the BTSs measured by the navigator is given in

Fig. 9. The mapper measured similar carrier-to-noise values.

0 20 40 60 80 100 120 140 160 180

20

30

40

50

60

70

Fig. 9. Carrier-to-noise ratios {C/N0n}9

n=1of all the cellular BTSs

measured by the navigator. The carrier-to-noise ratios measured by the mapperwere of similar values.

The CD-cellular measurements were used to estimate

the navigating receiver’s trajectory via the mapper/navigator

framework developed in Section III. The experimental setup,

the SOP BTS layout, and the true and estimated navigator

UAV trajectories are shown in Fig. 10. The position RMSE

was found to be 63.1 cm. Note that the Least-Squares AM-

Biguity Decorrelation Adjustment (LAMBDA) method [33]

implemented at the Delft University of Technology was used

to solve for the integer ambiguities [35].

Trajectories

UAV's Navigation System

CDMA (with Mapper)

Ettus E312USRP

CDMA Antenna

GPS Antenna

Navigator

Mapper

BTS 1

BTS 5

BTS 4

BTS 6

BTS 7

BTS 3

BTS 2

BTS 8

BTS 9

Position RMSE: 63.06 cm

Total Traversed Trajecory: 1.72 Km

1 Km

Fig. 10. Experimental setup, the SOP BTS layout, and the true andestimated navigator UAV trajectories via CD-cellular measurements in themapper/navigator framework. The true and estimated trajectories are shownin solid and dashed lines, respectively. Map data: Google Earth.

B. Single UAV Navigation Results with Precise Cellular Car-

rier Phase Measurements

Two experiments were conducted at different times. In the

first experiment, the same setup described in Subsection VII-A

was used, except that the navigator was navigating without

the mapper and was employing the framework developed in

Section IV. In the second experiment, a DJI Matrice 600

was equipped with the same hardware described in Subsection

VII-A and the on-board USRP was tuned to the same carrier

frequency. The cellular carrier phase measurements were also

given at a rate of 37.5 Hz, i.e., T = 0.0267 ms. The ground-

truth reference for the UAV trajectory was taken from its on-

board navigation system, which also uses GPS, an IMU, and

other sensors. The experimental setup and SOP BTS layout

for the second experiment are shown in Fig. 11.

In both experiments, the UAVs had access to GPS for 10

seconds, then GPS was cut off. During the time where GPS

was available, the cellular signals were used to cluster the cel-

lular SOPs and characterize the clock deviations, as described

in Subsection V-B. In the first experiment, the UAV traversed

a trajectory of 1.72 Km, which was completed in 3 minutes.

The receiver was listening to the same 9 CDMA BTSs as in

Fig. 10, with the same carrier-to-noise ratios as in Fig. 9. The

navigation results are shown in Fig. 12. The optimal clustering

was found to be C1 = {BTS1 , BTS 5, BTS 7, BTS 8},

C2 = {BTS 2, BTS 3, BTS 6 }, and C3 = {BTS 4, BTS 9}.

The position RMSE was calculated to be 36.61 cm.

Cellular Antenna

Ettus E312 USRP

BTS 1

BTS 5

BTS 4

BTS 6

BTS 7

BTS 3

BTS 2

GPS Antenna

1 Km

Fig. 11. Experimental setup and the SOP BTS layout for the secondexperiment demonstrating a single UAV navigating with precise cellularcarrier phase measurements. Map data: Google Earth.

Trajectories

UAV's Navigation System

CDMA (without Mapper)

Position RMSE: 36.61 cm

Total Traversed Trajecory: 1.72 Km

Fig. 12. First experiment demonstrating a single UAV navigating with precisecellular carrier phase measurements. The true and estimated trajectories areshown in solid and dashed lines, respectively. Map data: Google Earth.

In the second experiment, the UAV traversed a trajectory of

3.07 Km completed in 325 seconds. The receiver was listening

to the 7 CDMA BTSs shown in Fig. 11. The carrier-to-noise

ratios of all the BTSs measured by the navigating UAV in

the second experiment are given in Fig. 13 and the navigation

results are shown in Fig. 14. The optimal clustering was found

to be C1 = {BTS 1, BTS 2, BTS 3, BTS 4, BTS 6} and

C2 = {BTS 5, BTS 7}. The position RMSE was calculated

to be 88.58 cm.

0 20 40 60 80 100 120 140 160 180

20

30

40

50

60

70

Fig. 13. Carrier-to-noise ratios of all {C/N0n}7

n=1the cellular BTSs

measured by the navigating UAV for the second experiment.

The experimental results are summarized in Table I.

UAV's Navigation System

CDMA (without Mapper)

Position RMSE: 88.58 cm

Total Traversed Trajectory: 3.07 Km

Trajectories

Fig. 14. Second experiment demonstrating a single UAV navigating with pre-cise cellular carrier phase measurements. The true and estimated trajectoriesare shown in solid and dashed lines, respectively. Map data: Google Earth.

TABLE IEXPERIMENTAL RESULTS

FrameworkExperiment 1

RMSE [cm]

Experiment 2

RMSE [cm]

CD-Cellular withMapper/Navigator

63.06 –

Single UAV 36.61 88.58

C. Discussion

First, it is important to note that all RMSEs were calculated

with respect to the trajectory returned by the UAVs’ on-board

navigation system. Although these systems use multiple sen-

sors for navigation, they are not equipped with high precision

GPS receivers, e.g., Real Time Kinematic (RTK) systems.

Therefore, some errors are expected in what is considered to

be “true” trajectories taken from the on-board sensors. The

hovering horizontal precision of the UAVs are reported to be

2 meters for the X-Star Premium by Autel Robotics and 1.5

meters for the Matrice 600 by DJI.

Second, it can be noted that the CD-cellular with map-

per/navigator framework under-performed compared to the

single UAV framework. This can be due to: (1) poor synchro-

nization between the mapper’s and navigator measurements

and (2) errors in the mapper position. It is important to note

that the mapper was mobile during the experiment and the

position returned by its on-board navigation system was used

as ground-truth. Consequently, any errors in the GPS solution

would have degraded the navigator’s estimate.

Third, the RMSEs reported in this section are for optimal

clustering. In the 10 seconds during which GPS was available,

a search was performed to optimally cluster the clock biases

using the rule-of-thumb discussed in Subsection VI-C. The

search took less than 3 seconds. The RMSEs without cluster-

ing (only one bias is estimated) are 48 cm and 97 cm for the

first and second experiments, respectively.

VIII. CONCLUSION

This paper presented two frameworks for precise UAV

navigation with cellular carrier phase measurements. The first

framework relies on a mapping UAV and a navigating UAV.

Both UAVs are making carrier phase measurements to the

same cellular SOPs and share these estimates to produce

the double difference carrier phase measurement. During the

period when GPS is available, the UAVs use their carrier phase

measurements and known position to estimate the integer

ambiguities. Once GPS is cutoff, the navigator can produce an

estimate of its position. Experimental results showed a 63.06

cm position RMSE with this framework.

The second framework leverages the relative stability of

cellular BTSs clocks. This stability also allows to parameterize

the SOP clock biases by a common term plus some small

deviations from the common term, which alleviates the need

for a mapper. The clock deviations were subsequently mod-

eled as a stochastic sequence using experimental data. Next,

performance bounds were established for this framework.

Experimental data show that a single UAV can navigate with

sub-meter level accuracy for more than 5 minutes using this

framework, with one experiment showing 36.61 cm position

RMSE and another showing 88.58 cm position RMSE.

ACKNOWLEDGMENT

The authors would like to thank Joshua Morales for his help

in data collection.

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